For example, how could you trust a bank to keep currency for you, if there is no way to prove who you are when you return? What if the people you dealt with died? On the other hand, how could a bank give you your money, not knowing if you were telling the truth or not? You could even hire people to vouch for you.
To kind of expand on the question, to what extent would a photo ID have been useful or necessary before the 20th century? I'd imagine that most people would be interacting with a relatively limited group of people, but obviously that wouldn't always be the case.
Also follow up when did letters of introduction stop being widespread?
Not to discourage any discussion, because the things I've seen so far mostly only relate to forms of ID, and how one might identify themselves in specific instances or time periods, but this answer might help give you a clue
There's also a FAQ section for identification
Essentially, communities were much more closely knit. You'd know your neighbors much more, the people there could vouch for you, you don't have a thousand options for banks or shops, if it's a place you require the services of they likely deal with you on a regular enough basis that You'd know you could trust the bank because it'd be the major local bank. They'd have a reputation the same way any place now still has a reputation. You wouldn't just deal with one person all the time every time, you'd be a pretty recognizable face. Even today, if there's a place you're a regular customer at, people will recognize you.